$$L^{2}$$-rigidity in von Neumann algebras.(English)Zbl 1170.46053

Let $$N$$ be a finite von Neumann algebra with trace $$\tau$$. If $$M$$ is a finite von Neumann algebra with trace $$\tau'$$ such that $$N\subset M$$ and $$\tau'|_M=\tau$$, and $$\delta$$ is a densely defined real closable derivation on $$M$$ into $$(L^2(M,\tau')\otimes L^2(M,\tau'))^\infty$$, then the associated deformation $$\{\eta_\alpha\}_\alpha$$ coming from resolvent maps is called an $$L^2$$-deformation for $$N$$.
If $$B\subset N$$ is a von Neumann subalgebra, then the inclusion of $$B$$ into $$N$$ is called $$L^2$$-rigid if any $$L^2$$-deformation for $$N$$ converges uniformly on the unit ball of $$B$$. $$N$$ is $$L^2$$-rigid if the inclusion $$N\subset N$$ is $$L^2$$-rigid. This property generalizes property (T), which can be viewed as an analogue for the vanishing of 1-cohomology into the left regular representation of a group.
It is proved that, if $$N$$ is a II$${}_1$$ factor which is not prime or has property $$\Gamma$$ of Murray and von Neumann, then $$N$$ is $$L^2$$-rigid.
As a consequence, it is shown that, if $$M$$ is a free product of diffuse von Neumann algebras, or if $$M=LG$$, where $$G$$ is a finitely generated group with $$\beta_1^{(2)}(\Gamma)>0$$, then any non-amenable regular subfactor of $$M$$ is prime and doesn’t have properties $$\Gamma$$ or (T).

MSC:

 46L10 General theory of von Neumann algebras

Keywords:

rigidity; finite von Neumann algebra
Full Text:

References:

 [1] Bekka, M.E.B., Valette, A.: Group cohomology, harmonic functions and the first L 2-Betti number. Potential Anal. 6, 313–326 (1997) · Zbl 0882.22013 [2] Cipriani, F., Sauvageot, J.-L.: Derivations as square roots of Dirichlet forms. J. Funct. Anal. 201, 78–120 (2003) · Zbl 1032.46084 [3] Connes, A.: Classification of injective factors. Ann. Math. 104, 73–115 (1976) · Zbl 0343.46042 [4] Connes, A.: A type II1 factor with countable fundamental group. J. Oper. Theory 4, 151–153 (1980) · Zbl 0455.46056 [5] Connes, A.: Classification des facteurs. Proc. Symp. Pure Math. 38, 43–109 (1982) · Zbl 0503.46043 [6] Connes, A., Jones, V.F.R.: Property (T) for von Neumann algebras. Bull. Lond. Math. Soc. 17, 57–62 (1985) · Zbl 1190.46047 [7] Connes, A., Shlyakhtenko, D.: L 2-homology for von Neumann algebras. J. Reine Angew. Math. 586, 125–168 (2005) · Zbl 1083.46034 [8] Davies, E.B., Lindsay, J.M.: Non-commutative symmetric Markov semigroups. Math. Z. 210, 379–411 (1992) · Zbl 0761.46051 [9] Ge, L.: Applications of free entropy to finite von Neumann algebras. II. Ann. Math. (2) 147(1), 143–157 (1998) · Zbl 0924.46050 [10] Haagerup, U.: An example of a nonnuclear C *-algebra, which has the metric approximation property. Invent. Math. 50, 279–293 (1979) · Zbl 0408.46046 [11] Haagerup, U.: Injectivity and Decomposition of Completely Bounded Maps. Lect. Notes Math., vol. 1132, pp. 170–222. Springer, Berlin (1985) · Zbl 0591.46050 [12] Ioana, A., Peterson, J., Popa, S.: Amalgamated free products of w-rigid factors and calculation of their symmetry groups. Acta Math. 200(1), 85–153 (2008) · Zbl 1149.46047 [13] Jung, K.: Strongly 1-bounded von Neumann algebras. Geom. Funct. Anal. 17(4), 1180–1200 (2007) · Zbl 1146.46034 [14] Ma, Z.-M., Röckner, M.: Introduction to the Theory of (Non-Symmetric) Dirichlet Forms. Universitext. Springer, Berlin (1992) · Zbl 0826.31001 [15] Martin, F., Valette, A.: On the first L p -cohomology of discrete groups. Groups Geom. Dyn. 1(1), 81–100 (2007) · Zbl 1175.20045 [16] Murray, F.J., von Neumann, J.: On rings of operators IV. Ann. Math. (2) 44, 716–808 (1943) · Zbl 0060.26903 [17] Ozawa, N.: Solid von Neumann algebras. Acta Math. 192(1), 111–117 (2004) · Zbl 1072.46040 [18] Ozawa, N.: A Kurosh type theorem for type II1 factors. Int. Math. Res. Not., Art. ID 97560, 21 pp. (2006) · Zbl 1114.46041 [19] Ozawa, N., Popa, S.: Some prime factorization results for II1 factors. Invent. Math. 156, 223–234 (2004) · Zbl 1060.46044 [20] Peterson, J.: A 1-cohomology characterization of property (T) in von Neumann algebras. Preprint (2004). math.OA/0409527 [21] Popa, S.: Orthogonal pairs of *-subalgebras in finite von Neumann algebras. J. Oper. Theory 9(2), 253–268 (1983) · Zbl 0521.46048 [22] Popa, S.: Correspondences. INCREST Preprint (1986). unpublished [23] Popa, S.: Some rigidity results for non-commutative Bernoulli shifts. J. Funct. Anal. 230(2), 273–328 (2006) · Zbl 1097.46045 [24] Popa, S.: On a class of type II1 factors with Betti numbers invariants. Ann. Math. (2) 163(3), 809–899 (2006) · Zbl 1120.46045 [25] Popa, S.: Strong rigidity of II1 factors arising from malleable actions of w-rigid groups I, II. Invent. Math. 165(2), 369–408, 409–451 (2006) · Zbl 1120.46043 [26] Popa, S.: On Ozawa’s property for free group factors. Int. Math. Res. Not., 11, Art. ID rnm036, 10 pp. (2007) · Zbl 1134.46039 [27] Sauvageot, J.-L.: Tangent bimodules and locality for dissipative operators on C *-algebras. In: Quantum Probability and Applications, IV. Lect. Notes Math., vol. 1396, pp. 322–338. Springer, Berlin (1989) · Zbl 0682.46042 [28] Sauvageot, J.-L.: Quantum Dirichlet forms, differential calculus and semigroups. In: Quantum Probability and Applications, V. Lect. Notes Math., vol. 1442, pp. 334–346. Springer, Berlin (1990) [29] Sauvageot, J.-L.: Strong Feller semigroups on C *-algebras. J. Oper. Theory 42, 83–102 (1999) · Zbl 0998.46039 [30] Thom, A.: L 2-cohomology for von Neumann algebras. Geom. Funct. Anal. 18, 251–270 (2008) · Zbl 1146.46035 [31] Voiculescu, D.: The analogues of entropy and of Fisher’s information measure in free probability theory, V. Invent. Math. 132, 189–227 (1998) · Zbl 0930.46053
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.